Gaussian Elimination and Gauss Jordan Elimination (Gauss Elimination Method)

in this video I want to talk about gauss-jordan elimination and Gauss regular Gaussian elimination and how to do both of those and what the differences are so both of these are used to solve systems of linear equations they're basically just an easier way to do it than what you learned and say like algebra 1 or algebra 2 or something like that so here I have an example I've got three equations I've got X plus y plus 2z equals 9 2x plus 4y minus 3z equals 1 and 3x plus 6y minus 5z is 0 and the first thing we're gonna do is we're gonna put these in a matrix so I take the coefficients of all these numbers so the first column is going to be the X column and the coefficient of these are 1 2 & 3 my second column will be the Y's so I take all the coefficients of the Y's that's 1 4 and 6 the third column will be the Z's that's 2 negative 3 negative 5 and then I draw this bar that's kind of like the equals and here I put the constants 9 1 and 0 so this is called my Augmented matrix I'm gonna really need this anymore and now I'm gonna use we're called row operations so basically I can add rows to other rows or I can subtract rows to other rows or I can multiply rows by a constant or a fraction so those are the basic operations and I'm going to be using and my goal for the Gaussian elimination is to get a 1 and then 0 is underneath it so my I'm gonna have a leading one which is already what I have here and then 0 is underneath okay and then we'll see what happens after that so right here I already have one so there's nothing to do but I want zeros underneath it so I need to add my first row in some way so that I'll get a zero underneath it since this is a one I need to get it to to a two and in fact I need to get rid of this too I'm gonna do minus two times Row 1 plus Row 2 becomes my new Row 2 so this is the notation that you're going to use so this says I have minus twice Row 1 and I'm adding that to Row 2 and that becomes my new Row 2 all right so minus 2 times Row 1 that would be minus 2 minus 2 minus 4 minus 18 so I'm just taking minus 2 times each of these numbers and then if I add these correspondingly we'll see what happens so I've got minus 2 plus 2 is 0 I've got minus 2 plus 4 is 2 I've got minus 4 minus 3 is minus 7 and I've got minus 18 plus 1 is minus 17 so that was my first operation so I did minus 2 times the first row I added it to the second row so I've accomplished part of my goal and getting zeros underneath this one but I want zeros all the way down so basically I want this 3 to turn into a 0 can you guess how I might do that well I want to take my first row and this time I'll multiply it by a minus 3 so I can add it to my third row and that will cancel out this 3 so that's how I'm picking these numbers so I can get rid of the numbers right underneath so if I take minus the first row times 3 plus the third row becomes my new third row so what's minus 3 Row 1 well that'll be minus 3 minus 3 2 times minus 3 - 6 - 3 x 9s - 27 and now I'm gonna add this to the third row I'm gonna add the components together so have -3 plus 3 that's 0 I have minus 6 plus website - 3 plus 6 is positive 3 they've got minus 6 minus 5 that's minus 11 and I have minus 27 plus 0 minus 27 so that's my first step and now I've accomplished my goal of having zeros below this leading one so now what I do now that this Row is done I moved to the second row and now I want my leading one again well I've got a 0 so that's fine where it is we already said that was fine but now here's a 2 I want this to be a 1 I want this to to not be there I want a 1 and then 0 is underneath so I'm gonna take this row and I'm gonna multiply it by 1/2 to get rid of that - so I take 1/2 of Row 2 and that becomes my new Row 2 in other words I'm just dividing each term by 2 so that's 2 divided by 2 is 1 minus 7 divided by 2 is minus 7 halves - 17 divided by 2 is minus 17 over 2 and now what do we do well I want a 0 right underneath this so I need to take minus 3 times Row 2 - 3 times Row 2 plus Row 3 becomes my new Row 3 notice how I'm using the row right above it to get rid of the zeros below so let's look at this it's gonna get a little involved but I'm sure you'll be alright so minus 3 times 0 is 0 I'm not even gonna bother writing it - 3 times 1 is minus the - three times - seven halves well - a minus is a plus three times seven is twenty-one halves over here - uh - is a plus 3 times 17 let's see 3 times 10 is 30 3 times 7 is 21 I think it's gonna be 51 halves ugly number sorry and now it's gonna happen if I add these component wise I have 0 plus 0 is 0 minus 3 plus 3 is 0 and maybe I should change these 2 over 2 so I can use these fractions so minus 11 is 22 over 2 minus 27 is minus 54 over 2 so I've got 21 over 2 minus 22 over 2 that's minus 1 over 2 and I've got 51 over two minus 54 over 2 that's minus 3 over 2 and now I've accomplished my goal of a leading one with zeros underneath now I go to my very last row 0 0 there good I want this to be a 1 so what do I have to do to get rid of a 1/2 how do I make negative 1/2 1 well if I do minus 2 times Row 3 make that my new Row 3 that should do the trick so if I multiply this by minus 2 minus 2 times minus 1/2 is 1 minus three-halves times 2 is positive 3 and now I've got a one was ears underneath so now this what I've done here the leading ones with zeros underneath this is row echelon form okay I don't know why it's called that that's what it's called and this is as far as you would go with Gaussian elimination you would put it in row echelon form and you could solve your system like this so here's how we would solve the system I look here I basically have 0 x + 0 y + 1 z equals 3 and this simply means that Z equals 3 I can just read the answer I've got 1 Z equals 3 let me look at my second equation I've got 0 X plus 1 Y minus 7 halves Z equals minus 17 halves but I just found out that Z was 3 so let me plug 3 in 4 z that's Y minus 7 halves times 3 equals minus 17 halves that's why 3 times 7 is 21 halves equals minus 17 halves and now if I add 21 halves over minus 17 plus 21 is 4 halves or 2 so I know that y equals 2 and then I look at my very first equation I've got X 1 X plus 1 y plus 2z equals 9 but I know what Y is y is 2 its 2z I know what Z is Z is 3 which means that x equals let's see I've got 9 minus 2 times 3 is 6 minus 2 it's gonna be 3 minus 2 I think that's gonna be 1 yes okay and there you have it you've solved the system of equations using Gaussian elimination so you see how that works we got these leading ones we got rid of everything underneath so there was eros and the leading ones and then I did this back substitution thing I started from the bottom and I substituted I solved the second equation and I substituted to get the first equation that's Gaussian elimination but if you want to do gauss-jordan elimination we want to put it in reduced row-echelon form so this is row epsilon form Gauss Jordan is reduced row echelon form and what that is is that's just ones along the diagonal and zeros everywhere else so I'm kind of kind of do the same thing but I'm just gonna be using the matrix I'm not going to be doing using equations and substituting so just like I got zeros underneath the ones now I want zeros above the ones as well so now I start with my third row and I think what can i how can I add my third rode to my second row so that there's a zero right here well since this is already a one if I take seven halves times my Row three and I add it to my Row two to make that my new Row two that should cancel the thing out right above because I've got zero zero seven halves times one is seven halves and then seven halves times three is twenty-one halves and now I add this to my second row 0 plus 0 is 0 0 plus 1 is 1 7 halves minus 7 halves is 0 as desired and then 21 over 2 minus 17 over 2 that's 4 over 2 and we found that to be 2 and would you look at that I can already read that y equals 2 just like we found using Gaussian elimination and now I'm going to do almost the same thing I'm gonna have to use the third row into the second row and add it to the first row to get rid of the 0 or to get rid of this number and this number so to use the third row to get rid of this 2 since I want zeros all the way up I'm going to have to do two times Row three in fact minus two times Row three add it to Row one and that'll be my new Row 1 so minus two times my third row that's zero zero minus two minus two times 3 is minus six and now if I add this to my first row 0 plus 1 is 1 0 plus 1 is 1 minus 2 plus 2 is 0 and then minus 6 plus 9 is 3 we're almost done now that I have zeros completely above this one I just need a 0 above this one right here so I'm just gonna take minus Row 2 plus Row 1 to give me my new Row 1 and now I'm just adding the negative this to this so that's 0 minus 1 0 minus 2 if I add this to this 0 plus 1 is 1 minus 1 plus 1 is 0 0 plus 0 is 0 minus 2 plus 3 is 1 and now this is in reduced row echelon form and that's how you do Gauss Jordan elimination and you'll notice that we can just read off the answers like we got last time x equals 1 y equals 2 and Z equals 3 just as before so that's how you do Gaussian elimination and Gauss Jordan elimination just as a refresher Gaussian elimination you do row echelon form Gauss Jordan elimination is reduced row echelon form I hope you got something out of this video thank you so much for watching please subscribe for more videos like this and have a great day

Channel: BriTheMathGuy

Views: 261,618

Rating: 4.9089746 out of 5

Keywords: math, online math help, algebra, gaussian elimination, gauss jordan elimination, linear algebra, systems of linear equations, solving systems of linear equations, linear equations, matrix, matrices, row reduction

Id: AhUyh-2aPEc

Length: 15min 13sec (913 seconds)

Published: Thu Jun 09 2016